Algebra
# Roots of Unity

Given $z^2+z+1=0,$ find the value of

$\left(z+\dfrac{1}{z} \right)^2+\left(z^2+\dfrac{1}{z^2}\right)^2+\left(z^3+\dfrac{1}{z^3} \right)^2+\cdots+\left(z^{21}+\dfrac{1}{z^{21}}\right)^2.$

$\dfrac{31-2\delta_{1}}{1-2\delta_{1}} +\dfrac{31-2\delta_{2}}{1-2\delta_{2}}+\dfrac{31-2\delta_{3}}{1-2\delta_{3}}$

If $1,\delta_{1},\delta_{2},\delta_{3}$ are distinct fourth roots of unity, then evaluate the expression above.

If $w=e^{2\pi{i}/5}$, find $\sum_{k=1}^{4}\frac{1}{1+w^k+w^{2k}}$